Schreinemaker's Bundle

When a number of metamorphic phase reactions meet at a point, Schreinemaker analysis
enables us to say something about the order of the curves and even something about their
angles.

The first step is to list all the reactions whose curves meet at a point and enumerate
all the phases in all the reactions. If P is the number of phases in all the reactions
then:

P phases are present at the reaction point where all curves intersect, and F = 0.

P - 1 phases are present along each reaction curve, and F = 1.

P - 2 phases are present normally, and F = 2.

For example, consider these reactions:

Muscovite + Quartz == Orthoclase + Sillimanite + Water

Orthoclase + Quartz + Water === Silicate Melt

Muscovite + Quartz + Water === Sillimanite + Silicate Melt

Muscovite + Quartz === Orthoclase + Sillimanite + Silicate Melt

Muscovite + Silicate Melt === Orthoclase + Sillimanite + Water

Is the water liquid or vapor? Both, and neither. Since we have silicate melt,
we are way above the boiling point of water, but under pressure, boiling point
increases. If we follow the boiling curve of water, the liquid gets less dense
and the vapor gets more dense until they become indistinguishable. This point,
called the critical point, happens at 374 C and 220 bars for water, which
are geologically pretty moderate conditions. So in almost all metamorphic and
igneous settings, water is supercritical and can best be considered a very dense
vapor.

Altogether, six phases are involved. From here on, phases will be abbreviated as shown.

Muscovite (M)

Orthoclase (O)

Sillimanite (S)

Quartz (Q)

Silicate Melt (L = Liquid)

Water (W)

The fundamental principle is shown below:

First Step: Label Reactions

We label the reactions with the phase or phases that do not participate in the
reaction:

Muscovite + Quartz == Orthoclase + Sillimanite + Water (L)

Orthoclase + Quartz + Water === Silicate Melt (M, S)

Muscovite + Quartz + Water === Sillimanite + Silicate Melt (O)

Muscovite + Quartz === Orthoclase + Sillimanite + Silicate Melt (W)

Muscovite + Silicate Melt === Orthoclase + Sillimanite + Water (Q)

Second Step: Select a Curve and Determine which Sides Other Curves go on.

We'll begin with one curve, say:
Muscovite + Quartz == Orthoclase + Sillimanite + Water (L)
Label one side of the line with the reactants and the other with the products. For now, we
can pick the sides arbitrarily.

Here's the logic of labeling the curves as we did above. Consider the curve
labeled
M, for example. Along the M reaction curve, muscovite is not involved in any chemical
reactions. Obviously, then, this curve cannot be on the side of the (L) curve where
muscovite is involved in chemical reactions. It can't be on the reactant side of
the line (M+Q); it has to be on the product side (O+S+W). In general, labeled
curves have to go on the side of the line where the labeled phase is not
involved. If we label the
curve with reactants on one side and products on the other, then the labels for the
remaining curves go opposite the respective phases.

In this example, Muscovite and Quartz are involved as reactants above the line, so the
reaction curves where neither phase is involved cannot be on that side of the line. The M
and Q curves have to be on the opposite side of the line.

Note that we have one reaction:
Orthoclase + Quartz + Water === Silicate Melt (M, S)
that only involves only four phases.

Third Step: Begin Placing Remaining Curves

The fact that one curve has two labels is very significant. We can see that the two
labels lie on opposite sides of the (L) line. Thus, the (M,S) line simply crosses the (L)
line - it does not end at the intersection. Let's begin with this line. We can put the
labels on the proper sides of the (L) line as shown, but what about the reactants? The
liquid phase must occur on the opposite side of the (M,S) line from the (L) line. Thus we
have O+Q+W on the left and L on the right as shown below. For compactness, we'll omit the
plus signs: MQ means M + Q, and so on.

Note that we have no information about the angles between the
lines. Also we could just as easily have put the reactants on opposite sides of the (L)
line and gotten a mirror image diagram. What this method gives us is the topology
of the phase diagram - what lines intersect and in what relative order. Actual angles and
left- or right-handedness have to be derived from actual temperature and pressure data.

The
dashed line labeled L' is called the metastable extension of
line (L)

The metastable extension means this. The vertical line is the boundary where
the reaction O+Q+W = L occurs. Now suppose we don't have any excess silica. Then
the reaction can't go. That means we could still go to the right of the vertical
curve and have the reaction on the horizontal line occur. If we approach the
metastable extension from below, we would have O+S+W reacting to produce M+Q.
But now we have quartz present, so the reaction O+Q+W = L can go. There could
also be kinetic reasons why the reaction M+Q=O+S+W might go instead of some more
thermodynamically favorable reaction. The metastable extension will turn out to
be important, as we shall see below.

Let's consider another line, like (O). It has to occur above (L) and to the right of
(M,S) to be consistent with their labeling (at left, below). The reactants have to be
placed so that S and L are on the opposite side of (O) from (S) and (L), that is, with M +
Q + L on the left of (O) and S + L on the right (below, right). The metastable extension
of (O) is indicated by O'.

The real power of this method now becomes apparent.

For any given line, all the other line labels must be consistent with the
reaction represented by the line.

For a sector bounded by two lines, the line labels and the reactants within that field
must include all phases. The line labels cannot include phases present among the reactants
and vice versa. For example, in the diagram above, the sector enclosed by (L) and (S)
includes phases M, Q, O and W. The sector enclosed by (S) and (O) includes phases M, Q, L
and W.

A labeled reaction curve means that some phase does not participate in any
reaction. Thus, the curve must be opposite all reactions where that phase
participates. The metastable continuation of each reaction curve, on the
other hand, must extend into a sector where the
phase represented by the curve label is a reactant. For example, O' above extends into a
sector where O is a prticipant in both reactions. This is a powerful constraint on angles.
The labeling rules allow us to specify the order of the curves around an intersection. The
metastable continuation rule implies we are not completely free to change angles
arbitrarily.